The logistic-normal integral and its generalizations

We consider the solutions of the one-dimensional heat equation in an unbounded domain with initial conditions of the form f (x)/(1 + exp(sigma x)). This includes as a particular case the logistic-normal integral, which corresponds to f (x) = 1. Such initial conditions appear in stochastic calculus problems, and the numerical simulation of short-rate interest rate models and credit models with log-normally distributed short rates and hazard rates respectively. We show that the solutions at time t can be computed exactly on a grid of equidistant points of width sigma t in terms of the solutions of the heat equation with initial condition f (x). The exact results on the grid can be used as nodes for a precise interpolation. Series representation of the solutions can be obtained by an application of the Poisson summation formula. (c) 2012 Elsevier B.V. All rights reserved.